What is the range of f(x)=(4e^x)/(1+2e^x) ?

The range of f(x)=(4e^x)/(1+2e^x) is 0<f(x)<2

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range of f(x)=(4e^x)/(1+2e^x)

range

f (x) = \frac{4 e ^{x}}{1 + 2 e ^{x}}

0 < f (x) < 2

Interval Notation

(0, 2)

Solution steps

The function range is the combined domain of the inverse functions

Find the inverse function(s) of:

\frac{4 e ^{x}}{1 + 2 e ^{x}}

Inverse of

\frac{4 e ^{x}}{1 + 2 e ^{x}} : ln (- \frac{x}{2 ( x - 2 )})

ln (- \frac{x}{2 ( x - 2 )})

Find the domain of each inverse function

Domain of

ln (- \frac{x}{2 ( x - 2 )}) : 0 < x < 2

Combine the intervals

0 < f (x) < 2

What is the range of f(x)=(4e^x)/(1+2e^x) ?
The range of f(x)=(4e^x)/(1+2e^x) is 0<f(x)<2